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A Hybrid Relaxation-Heuristic Framework for Solving MIP with Binary Variables

Zayn Wang

TL;DR

This work tackles the NP-hard problem of MIQP with binary variables and cardinality constraints, focusing on portfolio optimization. It introduces a hybrid framework that first builds a diverse initial solution pool from three exact relaxations—Line Model, Dual Model, and Augm Model—and then refines candidates with a Genetic Algorithm and Neighborhood Search to produce high-quality binary decisions. Empirical results on OR Library portfolio datasets show state-of-the-art performance, with solutions closely tracking the unconstrained efficient frontier and achieving favorable objective gaps compared to the Gurobi baseline. The approach demonstrates robustness and scalability by blending exact relaxation insights with heuristic search, offering a path to extend to broader MIP problems.

Abstract

Mixed-Integer Programming (MIP), particularly Mixed-Integer Linear Programming (MILP) and Mixed-Integer Quadratic Programming (MIQP), has found extensive applications in domains such as portfolio optimization and network flow control, which inclusion of integer variables or cardinality constraints renders these problems NP-hard, posing significant computational challenges. While traditional approaches have explored approximation methods like heuristics and relaxation techniques (e.g. Lagrangian dual relaxation), the integration of these strategies within a unified hybrid framework remains underexplored. In this paper, we propose a generalized hybrid framework to address MIQP problems with binary variables, which consists of two phases: (1) a Mixed Relaxation Phase, which employs Linear Relaxation, Duality Relaxation, and Augmented Relaxation with randomized sampling to generate a diverse pre-solution pool, and (2) a Heuristic Optimization Phase, which refines the pool using Genetic Algorithms and Variable Neighborhood Search (VNS) to approximate binary solutions effectively. Becuase of the page limit, we will only detailedly evaluate the proposed framework on portfolio optimization problems using benchmark datasets from the OR Library, where the experimental results demonstrate state-of-the-art performance, highlighting the framework's ability to solve larger and more complex MIP problems efficiently. This study offers a robust and flexible methodology that bridges relaxation techniques and heuristic optimization, advancing the practical solvability of challenging MIP problems.

A Hybrid Relaxation-Heuristic Framework for Solving MIP with Binary Variables

TL;DR

This work tackles the NP-hard problem of MIQP with binary variables and cardinality constraints, focusing on portfolio optimization. It introduces a hybrid framework that first builds a diverse initial solution pool from three exact relaxations—Line Model, Dual Model, and Augm Model—and then refines candidates with a Genetic Algorithm and Neighborhood Search to produce high-quality binary decisions. Empirical results on OR Library portfolio datasets show state-of-the-art performance, with solutions closely tracking the unconstrained efficient frontier and achieving favorable objective gaps compared to the Gurobi baseline. The approach demonstrates robustness and scalability by blending exact relaxation insights with heuristic search, offering a path to extend to broader MIP problems.

Abstract

Mixed-Integer Programming (MIP), particularly Mixed-Integer Linear Programming (MILP) and Mixed-Integer Quadratic Programming (MIQP), has found extensive applications in domains such as portfolio optimization and network flow control, which inclusion of integer variables or cardinality constraints renders these problems NP-hard, posing significant computational challenges. While traditional approaches have explored approximation methods like heuristics and relaxation techniques (e.g. Lagrangian dual relaxation), the integration of these strategies within a unified hybrid framework remains underexplored. In this paper, we propose a generalized hybrid framework to address MIQP problems with binary variables, which consists of two phases: (1) a Mixed Relaxation Phase, which employs Linear Relaxation, Duality Relaxation, and Augmented Relaxation with randomized sampling to generate a diverse pre-solution pool, and (2) a Heuristic Optimization Phase, which refines the pool using Genetic Algorithms and Variable Neighborhood Search (VNS) to approximate binary solutions effectively. Becuase of the page limit, we will only detailedly evaluate the proposed framework on portfolio optimization problems using benchmark datasets from the OR Library, where the experimental results demonstrate state-of-the-art performance, highlighting the framework's ability to solve larger and more complex MIP problems efficiently. This study offers a robust and flexible methodology that bridges relaxation techniques and heuristic optimization, advancing the practical solvability of challenging MIP problems.
Paper Structure (20 sections, 5 theorems, 15 equations, 2 figures, 3 tables)

This paper contains 20 sections, 5 theorems, 15 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mixed-Integer Programming
  3. Portfolio Optimization Model
  4. Problem Formulation
  5. Primal Model (Prim Model)
  6. Linear Model (Line Model)
  7. Duality Model (Dual Model)
  8. Augmented Model (Augm Model)
  9. Proposed Solution Approach
  10. Initial Solution Pool
  11. Genetic Algorithm
  12. Neighborhood Search Algorithm
  13. Experiment Results
  14. Dataset Description
  15. Environment Description
  16. ...and 5 more sections

Key Result

Theorem 2.1

For all Primal Model, or original cardinality binary constrained optimization problem, can be defined as: where (prim1) reveals the basic LP or QP object to be minimized, and if it is a MILP, then $Q = \mathbf{0}$; (prim2) reveals the basic LP or QP constraints; (prim3) reveals the upper bound and lower bound for $\mathbf{x}$; (prim4) reveals the cardinality constraints; and (prim5) reveals the i

Figures (2)

  • Figure 1: A schematic visualizing our approach. Supported by Open-Source Excalidraw.
  • Figure 2: A schematic visualizing the objective gaps. Supported by Matplotlib.

Theorems & Definitions (10)

  • Definition 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • Definition 2.7
  • Theorem 2.8